Refers to special locations inside the core of a micro(processor|controller). These can be thought of as memory locations, but they are so much more.

Much

These suckers can be accessed within the same clock cycle they are being used. Usually you have to go out to main memory, pluck the value from there, operate on it, and then put it somewhere. This can take forever! But write something to a register - BAM! It's there! Increment it - BAM! it's incremented. Some processors can operate on multiple registers in one machine cycle. Yowza!

These are used for ultra-tight and precise loops and algorithms, and are a basic way of passing information from one function to another. Part of a processor's bit rating (ie, a 32 bit processor) is determined by how large these registers are. You could have a 32 bit core, but if your registers are all 8bit then many people consider the processor to be an 8 bit (and, for all intents and purposes, it is as effective as an 8 bit)

Most people don't even think about the registers anymore. The OS or the compiler determines how they get used, and that's the end of it - which is ok, since you'll rarely run across someone who programs in a low enough language who touches the registers.